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1 AIAA-24-5 Computational Investigation of Airfoils with Miniature Trailing Edge Control Surfaces Hak-Tae Lee, Ilan M. Kroo Stanford University, Stanford, CA 9435 Abstract Miniature trailing edge effectors (MiTEs) are small flaps (typically % to 5% chord) actuated with deflection angles of up to 9 degrees. The small size, combined with little required power and good control authority, enables the device to be used for high bandwidth control. Recently, there have been attempts to use MiTEs as aeroelastic control devices, mainly to stabilize a wing operating beyond its flutter speed. However, the detailed aerodynamic characteristics of these devices are relatively unknown. The present study investigates the steady and unsteady aerodynamics of MiTEs. In order to understand the flow structure and establish a parametric database, steady state incompressible Navier-Stokes computations are performed on MiTEs with various geometries using INS2D flow solver. In addition, to resolve the dynamic characteristics, time accurate computation is implemented. Introduction The Gurney flap is a small flap used to increase the lift of a wing. It was developed and applied to race cars by Robert Liebeck and Dan Gurney in the 96 s. Numerous wind-tunnel tests and numerical computations have been performed on airfoils with Gurney flaps., 3 5, 8, 9 These studies confirm that despite their small size, Gurney flaps can significantly increase the maximum lift or the lift produced at a given angle of attack. The aerodynamic force alteration is produced by a small region of separated flow directly upstream of the flap, with two counterrotating vortices downstream of the flap effectively modifying the trailing edge Kutta condition. This Doctoral Candidate, Department of Aeronautics and Astronautics, AIAA Student Member Professor, Department of Aeronautics and Astronautics, AIAA Fellow Copyright c 24 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. mechanism was first proposed by Liebeck and later verified via flow visualization 8 and CFD 3 simulations. Miniature Trailing edge Effectors (MiTEs) are small movable control surfaces similar to Gurney flaps, at or near the trailing edge. MiTEs are deflected to large angles to produce control forces and moments that may be used for general flight control or aeroelastic control. Recently, Lee and Bieniawski 2 designed an aeroelastic control system to suppress flutter using a simple linear aerodynamic model of MiTEs. However, the experiments done by Solovitz and Bieniawski 2 suggest that significant nonlinear characteristics such as vortex shedding exists in the aerodynamics of MiTEs and more sophisticated aerodynamic models are required for higher performance control. Most of the previous work on Gurney flaps 3 6, 8 has concentrated on studying lift and drag, while varying the size of the flap and the angle of attack. As a control device, the focus of the current study is on the change in lift, drag, and pitching moment with fully deployed MiTEs as compared to the clean configuration. A blunt trailing edge is needed to provide a space behind the trailing edge to store the flap. For the present study, which involves a sliding rectangular plate behind the trailing edge, a blunt trailing edge with the thickness at least the same as the flap height is required. For steady state computations, the height, h, of the flap is varied for both sharp and blunt trailing edge airfoil as well as Reynolds number and angle of attack. For unsteady computations, time accurate simulation is performed for a fully deployed flap with the flow started impulsively. With this fixed grid time accurate computation, transient force response and vortex shedding frequencies are investigated. Time history of lift and moment coefficients are then computed with the flap sliding up and down in a harmonic motion over a range of frequencies. The latter results are obtained with moving grid computations.

2 h f = h w f t TE h f h l TE w f Figure 2: C-Grid used for sharp NACA2 airfoil Figure : Geometry of MiTEs attached to sharp and blunt trailing edge airfoils Flow Solver A two dimensional Reynolds-averaged Incompressible Navier-Stokes code, INS2D 2 is used. INS2D utilizes an artificial compressibility scheme that requires subiterations in the pseudo time domain to ensure a divergence free velocity field at the end of each physical time step. An upwind differencing scheme based upon flux-difference splitting is used for the convective terms, while a second-order central differencing is used for the viscous fluxes. The equations are solved using an implicit line relaxation scheme or generalized minimum residual method. The flow is assumed to be fully turbulent and the Spalart-Allmaras turbulence model is used, since this model is well known for its good performance in separated regions away from the wall. Geometry and Grid Figure shows the geometric parameters for the flaps attached to both sharp and blunt trailing edges. h f and w f are height and chordwise thickness of the flap respectively. For a blunt trailing edge airfoil, t T E is the trailing edge thickness and l T E is the overall projection thickness of the trailing edge including the flap. For both cases, h is the height of the flap measured from the airfoil surface. For a sharp trailing edge airfoil, h = h f = l T E, t T E =. () For a blunt trailing edge airfoil, t T E, l T E = t T E + h. (2) h is equal to h f for a sharp trailing edge airfoil but not necessarily the same as h f for a blunt trailing edge airfoil. In the later section, the significance of these geometric parameters, especially for the blunt trailing edge case, is explained. For the present study, NACA2 is chosen for the baseline airfoil and an airfoil with.5% thick trailing edge is constructed by linearly shearing the thickness distribution of the baseline airfoil using Equation 3. t blunt () = t sharp () +.5() (3) For sharp NACA2 airfoil, computations were performed using a single zone C-grid as shown in Figure 2. Far field boundaries are located 5c from the airfoil in upstream, downstream, top, and bottom where c is the chord length. Minimum grid size in the direction normal to the solid wal is set to. 5 c to ensure acceptable value of y +. The flaps are represented in the computational domain using the iblank function of INS2D. With iblank, any point in the grid can be specified as a no slip surface or blanked out to be a hole region. Figure 3 shows the grid around the trailing edge and the flap. Chordwise thickness of the flap, w f, is set to have the minimum possible value which is determined by the minimum grid point requirement for using iblank. 2

3 ξ η ξ η ξ η A B ξ η Figure 3: Grid near the sharp trailing edge N f Coarse Medium Fine 339 x (59+N f) 53 x (88+N f) 749 x (32+N f) C l C d C l C d C l C d C l C d Figure 4: Schematic diagram of G-Grid Table : C l and C d computed using grids of various resolution (R e =.5 6 ) A grid refinement study is performed on a baseline design of h =.5% at a zero angle of attack and a Reynolds number of.5 6 to find the proper number of grid points N ξ in ξ direction and N η in η direction as well as N f inside the flap region. Table shows the C l and C d values computed from twelve different grid resolutions. From the coarsest grid of 339 by 93 to the finest one of 749 by 25, the difference is less than %. Medium density with N f = 78 is selected as the baseline resolution for steady state computations. For a flap size other than.5%, N f is scaled linearly according to the flap height, h. To reduce computational time, all time accurate computations are completed with the coarse resolution and a baseline N f of 52 for the.5% flap. A new gridding scheme called G-grid is devised to represent a blunt trailing edge airfoil with MiTEs. A G-grid is similar to a C-grid but, η = grid line meets part of ξ = grid line at the wake cut surface which is perpendicular to the chord line. Figure 4 shows the schematic diagram of a G-grid. The boundary condition for MiTEs can be easily implemented by specifying no slip wall condition to both A and B segment. Figure 5: The actual G-Grid used for blunt trailing edge airfoil Figure 5 shows the actual G-grid used for the computations and Figure 6 shows the detailed grid structure around the trailing edge. Mesh resolution for this G-grid is chosen to be similar to that of C-grid and ranges from 387 by 27 for.5% flap to 553 by 257 for 3.% flap. A three-zone overset grid is used for the moving grid computations. As can be seen from Figure 7, zone I is a C-grid surrounding only the airfoil without the wake. Zone II is a rectangular region downstream of the trailing edge and contains the flap where the grid points on the flap surface are specified as solid wall boundaries and the points inside the flap are blanked out using iblank. Zone III is a small rectangular grid needed to define the solid 3

4 Zone I Leak blocking tap Zone III Solid walls Figure 6: Grid near the blunt trailing edge Zone I Airfoil Zone III Zone II Figure 7: Three zone overset grid used for moving flap computation wall for the blunt trailing edge. Detailed view near the trailing edge is given in Figure 8. The boundary values are updated from linear interpolation between zone I and II, and zone III and II. Note that the interfaces between zone I and III are solid wall boundaries and these two small taps block the flow between the trailing edge and the flap. Zone II slides up and down as a rigid body translation according to the motion of the flap. Grids are generated at each time step as well as the interface file that gives the information for updating the boundaries. Steady State Computation Results Steady state force coefficients were computed for various configurations. For a sharp NACA2 airfoil, the flap size, h, ranges from.5% to 3.% and for blunt NACA2 airfoil with trailing edge thickness of.5%, h ranges from.5% to 2.5%. The computations were also completed at three different Reynolds Zone I Moving flap Zone II Figure 8: Enlarged view of the overset grid near the trailing edge Airfoil NACA2 TE thickness, t T E (%).,.5 Flap size, h (%).5,.,.5, 2., 2.5, (3.) Reynolds number ( 6 ).,.5, 2. Angle of attack ( ) 5., 2.5,., 2.5, 5. Table 2: Conditions for the steady state computations numbers and five angles of attack summarized in Table 2. Figure 9 and Figure show the color maps of stagnation pressure and streamlines near the trailing edge for sharp and blunt NACA2 airfoils, respectively. As can be seen from both streamlines, the basic flow structures are the same. 4, 6, 8, Previous research suggests that the C l and C m remain more or less constant for moderate angles of attack. C l is plotted with respect to angle of attack, α, in Figure for both trailing edge thicknesses to closely examine the effect of angle of attack on C l. The Reynolds number is set to.5 6. C l increases as α increases for every flap size for both airfoils. The variations are, however, relatively minor compared to the values of C l. When the angle of attack is negative, the boundary layer thickens at the lower surface where the flap is attached and it is known that thick boundary layers reduce the effectiveness of Gurney flaps, thus reducing the C l. Conversely, thinner boundary layers enhanc the flap effectiveness at positive angles of attack. Figure also shows that C l is less sensitive to the angle of attack for the blunt trailing edge airfoil. For all the 4

5 h =.5% h =.% h =.5% h = 2.% h = 2.5% h = 3.% α (degree) (a) Sharp trailing edge Figure 9: Streamline and stagnation pressure map near the trailing edge of a NACA2 airfoil(h =.5%, α =, Re =.5 6 ). h =.5% h =.% h =.5% h = 2.% h = 2.5% α (degree) (b) Blunt trailing edge (t T E =.5%) Figure : C l with respect to α (R e =.5 6 ) subsequent results, zero angle of attack is assumed unless mentioned specifically. Figure : Streamline and stagnation pressure map near the trailing edge of a blunt NACA2 airfoil(t T E =.5%, h =.5%, α =, Re =.5 6 ) C l is plotted with respect to flap height, h, in Figure 2 for both airfoils and is compared to experimental data. C l increases monotonically 4, 6, 8 as h increases while the blunt trailing edge airfoil results closely follow the values from the sharp trailing edge ones. However the efficiency, defined as C l /(h/c), consistently decreases as h increases, as given in Figure 3. Although the blunt trailing edge results generally match the sharp one closely, the efficiencies for.5% and.% flap are notably higher than those for the sharp trailing edge. For the pitching moments, the ratio, C m / C l is nearly constant and close to 4, the value expected from the thin airfoil theory as the size of the flap approaches zero (Figure 4). The pressure profile given in Figure 5, computed at zero angle of attack to show the net effect of the flap, indicates that the increase in lift is relatively flat along the chord, which also confirms the trend for the relation between C m and C l. It is demonstrated throughout the results that the value of C l, and consequently the C m for both the sharp and blunt trailing edges match very well if the flap height is defined as the distance between the 5

6 .2.2 Thin airfoil theory Sharp trailng edge Blunt trailing edge.22 C m / (α= deg) Flap size (%) (a) Cm C l Thin airfoil theory Sharp trailng edge Blunt trailing edge with respect to h.4 Sharp trailing edge. Blunt trailing edge Jeffrey (Re = x e6, E423) Storms (Re = 2 x e6, NACA442) Moyse (Re = 2.2 x e6, NACA2) flap height, h (%) C m / Figure 2: Change in the C l at zero angle of attack with respect to flap height, h. Experimental data are plotted for comparison (Re =.5 6 ) Flap size (%, log scale) (b) Cm compared to the Thin airfoil C l theory Figure 4: C m C l with respect to h (Re =.5 6 ) /(h/c) (Re=.5 x 6, α= deg) Figure 3: height, h Sharp trailing edge Blunt trailing edge flap height, h (%) Flap efficiency, C l h/c with respect to flap airfoil surface and the end of the flap as indicated in Figure. This comparison suggests that the proper definition for the flap height should be measuring the distance from the airfoil surface rather than from any other reference line such as the chord line. Figure 6 illustrates the change in flap efficiency with respect to the Reynolds number for different flap heights. As can be seen, the efficiency monotonically increases with the Reynolds number regardless of the flap size although the variation is very small. As stated previously, smaller flaps have higher efficiency, but at the same time, Figure 6 shows that smaller flaps are more sensitive to the Reynolds number. Figure 7 provides the drag polar for a sharp NACA2 with.5% flap compared to the clean configuration. At low lift coefficients, the drag penalty of the flap is apparent, but at high C l values, the delay in the upper surface separation induced by the flap causes the airfoil with a flap to have lower drag. Drag increment, C d with respect to h is plotted at Figure 8 for a sharp NACA2 airfoil. C d increases steeply as h increases. At h = 3.%, C d is similar to the drag of clean NACA2 airfoil. 6

8 (a) U t c = (b) case case 2 case 3 t T E (%)...5 h (%) l T E (%) Re (based on l T E ) 22,5 45, 45, frequency, f Strouhal number, St Table 3: Summary of vortex shedding frequencies (c) U t c =.65 (d) C l Ut/c (e) U t c = (f) Figure 2: Time history, k = π/27.8 (g) U t c =.69 (i) U t c =.7 (k) U t c = (h) (j) (l) The frequency of this oscillation is about.5 Hz, which translates to the Strouhal number of.58 based on the height of the flap. For the flap height based Reynolds number of 22, 5, this Strouhal number agrees well with the experimental results. 8 To further investigate how the vortex shedding frequency changes with the geometry, time accurate computations were performed for a sharp trailing edge airfoil with a 3% flap and a.5% thick trailing edge airfoil with a.5% flap. As summarized in Table 3, the vortex shedding frequencies for both cases are the same, while this frequency is roughly half the value from the sharp trailing edge with a.5% flap. This result suggests that the proper characteristic length for the vortex shedding frequency should be defined as the distance between the bottom end of the flap and the upper end of the trailing edge, l T E, as seen in Figure. For the moving grid computation, trailing edge thickness of.% and flap height of.% were selected. All computations were completed at zero angle of attack and Reynolds number of.5 6 using a three-zone overset grid. Grid resolutions were set at 87 by 43 for zone I, 82 by 5 for zone II, and 9 by 3 for zone III. A time step of. was used Figure 2: Vortex shedding 8

9 C l.4 C l (a) Flap is near the neutral position Ut/c Ut/c Figure 22: Time history, enlarged view (a) U t c = 3. (b) U t c = C l.5 (c) U t c = 3.3 (d) U t c = Ut/c Figure 23: Time history, k = π/3 (b) Flap is near the top position to capture the vortex shedding and the grid file was generated at every time step to represent the flap sliding up and down in a harmonic motion. Figure 2 shows the time history of the lift coefficient for reduced frequency of k =.6. As can be seen from the enlarged view in Figure 22, at low reduced frequencies, the vortex shedding occurs constantly regardless of the position of the flap but the intensity is higher when the flap is nearly stationary at the top or bottom end. The vortex shedding frequency decreases as the flap moves away from the neutral position and increases as the flap moves towards the neutral position. For a high reduced frequency of k =.5 (Figure 23) a slight vortex shedding is observed only when the flap is at the up or down position where the velocity of the flap is close to zero. Figure 24 presents the sequence of streamlines and the stagnation pressure map of the flap moving from neutral to down position with k =.5. The magnitude and phase delay of the response at actuation frequency is computed at five reduced frequencies (k =.6,.75,.349,.524,.5). The resulting magnitude and phase delay are plotted with respect to reduced frequency in Figure 25. For comparison purpose, the magnitude and phase of the Magnitude (C l ) Phase (deg) (e) U t c = 3.6 (f) U t c = 3.75 Figure 24: Moving flap reduced frequency, k moving grid computation Theodorsen C(k) Figure 25: Frequency response compared to the magnitude and phase of the Theodorsen s function, C(k) 9

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